What is the relationship between median $M$, second quartile ${Q_2}$, ${5^{th}}$ decile ${D_5}$, and $5{0^{th}}$ percentile ${P_{50}}$ of a set of observations?
A. $M > {Q_2} > {D_5} > {P_{50}}$
B. $M = {Q_2} = {D_5} = {P_{50}}$
C. $M < {Q_2} < {D_5} < {P_{50}}$
D. $M < {Q_2}, {D_5} > {P_{50}}$

Hint: First, find the meaning of the median, second quartile, ${5^{th}}$ decile, and $50^{th}$ percentile. Then compare them to get the required answer.

Complete step by step solution:
Given, The median, second quartile, ${5^{th}}$ decile, and $50^{th}$ percentile of a set of observation are represented by $M$, ${Q_2}$, ${D_5}$, and ${P_{50}}$ respectively.
Let’s find out the relationship between the median, second quartile, ${5^{th}}$ decile, and $5{0^{th}}$ percentile of a set of observations.
We know that the median is the middle value of the data when the data is arranged in ascending order.
${2^{nd}}$ quartile: The ${2^{nd}}$ quartile splits the arranged data set in half. $50\% $ of the data lies below this point. So, it is called the median of the data set.
${5^{th}}$ decile: If an ordered data set is divided into $10$ parts, then cut points are called the decile. ${5^{th}}$ decile divides the arranged data set in half. Hence, it is called the median of the data set.
$50^{th}$ percentile: The $5{0^{th}}$ percentile is the second or median quartile of an arranged data set. It splits the data set in half. Hence, it is called the median of a data set.
Since the second quartile, ${5^{th}}$ decile and $50^{th}$ percentile of an arranged data set is called the median.
Therefore, the relationship between them is $M = {Q_2} = {D_5} = {P_{50}}$.

Option ‘B’ is correct

Note: To answer this problem we have to know about the quartile, decile, and percentile.
Quartile: The quartile is a division of the observations into four equal intervals based on the total number of observations.
Decile: The decile is a division of the observations into ten equal intervals based on the total number of observations.
Percentile: The decile is a division of the observations into hundred equal intervals based on the total number of observations.